This article provides partial solutions to Chinburg’s conjectures by studying a sequence of multivariate polynomials. These conjectures assert that for every odd quadratic Dirichlet character of conductor f, \(\chi _{-f}=\left( \frac{-f}{.}\right) \) , there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of \(L'(\chi _{-f},-1)\) . We prove that the Mahler measure of a polynomial family, denoted by \(P_d\) , can be expressed as a linear combination of the derivatives of Dirichlet L-functions. Specifically, this family provides solutions to the conjectures for conductors \(f=3,4,8,15,20\) , and 24. We further generalize Chinburg’s conjectures from real primitive odd Dirichlet characters to all primitive odd characters. For this generalized version, the polynomials \(P_d\) provide solutions for conductors 5, 7, and 9.
